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Mirrors > Home > MPE Home > Th. List > oppcinv | Structured version Visualization version GIF version |
Description: An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
oppcsect.b | ⊢ 𝐵 = (Base‘𝐶) |
oppcsect.o | ⊢ 𝑂 = (oppCat‘𝐶) |
oppcsect.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
oppcsect.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
oppcsect.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
oppcinv.s | ⊢ 𝐼 = (Inv‘𝐶) |
oppcinv.t | ⊢ 𝐽 = (Inv‘𝑂) |
Ref | Expression |
---|---|
oppcinv | ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4161 | . . 3 ⊢ ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)) = (◡(𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) | |
2 | oppcsect.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐶) | |
3 | oppcsect.o | . . . . . . 7 ⊢ 𝑂 = (oppCat‘𝐶) | |
4 | oppcsect.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | oppcsect.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | oppcsect.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | eqid 2820 | . . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
8 | eqid 2820 | . . . . . . 7 ⊢ (Sect‘𝑂) = (Sect‘𝑂) | |
9 | 2, 3, 4, 5, 6, 7, 8 | oppcsect2 17027 | . . . . . 6 ⊢ (𝜑 → (𝑌(Sect‘𝑂)𝑋) = ◡(𝑌(Sect‘𝐶)𝑋)) |
10 | 9 | cnveqd 5727 | . . . . 5 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = ◡◡(𝑌(Sect‘𝐶)𝑋)) |
11 | eqid 2820 | . . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
12 | eqid 2820 | . . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
13 | eqid 2820 | . . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
14 | 2, 11, 12, 13, 7, 4, 5, 6 | sectss 17000 | . . . . . . 7 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))) |
15 | relxp 5554 | . . . . . . 7 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) | |
16 | relss 5637 | . . . . . . 7 ⊢ ((𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → (Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → Rel (𝑌(Sect‘𝐶)𝑋))) | |
17 | 14, 15, 16 | mpisyl 21 | . . . . . 6 ⊢ (𝜑 → Rel (𝑌(Sect‘𝐶)𝑋)) |
18 | dfrel2 6027 | . . . . . 6 ⊢ (Rel (𝑌(Sect‘𝐶)𝑋) ↔ ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋)) | |
19 | 17, 18 | sylib 220 | . . . . 5 ⊢ (𝜑 → ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋)) |
20 | 10, 19 | eqtrd 2855 | . . . 4 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋)) |
21 | 2, 3, 4, 6, 5, 7, 8 | oppcsect2 17027 | . . . 4 ⊢ (𝜑 → (𝑋(Sect‘𝑂)𝑌) = ◡(𝑋(Sect‘𝐶)𝑌)) |
22 | 20, 21 | ineq12d 4173 | . . 3 ⊢ (𝜑 → (◡(𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌))) |
23 | 1, 22 | syl5eq 2867 | . 2 ⊢ (𝜑 → ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌))) |
24 | 3, 2 | oppcbas 16966 | . . 3 ⊢ 𝐵 = (Base‘𝑂) |
25 | oppcinv.t | . . 3 ⊢ 𝐽 = (Inv‘𝑂) | |
26 | 3 | oppccat 16970 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
27 | 4, 26 | syl 17 | . . 3 ⊢ (𝜑 → 𝑂 ∈ Cat) |
28 | 24, 25, 27, 6, 5, 8 | invfval 17007 | . 2 ⊢ (𝜑 → (𝑋𝐽𝑌) = ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋))) |
29 | oppcinv.s | . . 3 ⊢ 𝐼 = (Inv‘𝐶) | |
30 | 2, 29, 4, 5, 6, 7 | invfval 17007 | . 2 ⊢ (𝜑 → (𝑌𝐼𝑋) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌))) |
31 | 23, 28, 30 | 3eqtr4d 2865 | 1 ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∩ cin 3918 ⊆ wss 3919 × cxp 5534 ◡ccnv 5535 Rel wrel 5541 ‘cfv 6336 (class class class)co 7137 Basecbs 16461 Hom chom 16554 compcco 16555 Catccat 16913 Idccid 16914 oppCatcoppc 16959 Sectcsect 16992 Invcinv 16993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-tpos 7873 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-2 11682 df-3 11683 df-4 11684 df-5 11685 df-6 11686 df-7 11687 df-8 11688 df-9 11689 df-n0 11880 df-z 11964 df-dec 12081 df-ndx 16464 df-slot 16465 df-base 16467 df-sets 16468 df-hom 16567 df-cco 16568 df-cat 16917 df-cid 16918 df-oppc 16960 df-sect 16995 df-inv 16996 |
This theorem is referenced by: oppciso 17029 episect 17033 |
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